A simple inequality for the von Neumann–Jordan and James constants of a Banach space

Let CNJ(X) and J(X) be the von Neumann–Jordan and James constants of a Banach space X, respectively. We shall show that CNJ(X)⩽J(X), where equality holds if and only if X is not uniformly non-square. This answers affirmatively to the question in a recent paper by Alonso et al. [J. Alonso, P. Martín, P.L. Papini, Wheeling around von Neumann–Jordan constant in Banach spaces, Studia Math. 188 (2008) 135–150]. This inequality looks quite simple and covers all the preceding results. In particular this is much stronger than Maligranda's conjecture: .